document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=130497>Question 130497</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi there, was wondering if someone could explain this for me?<BR>\n" );
document.write( "Explain the meaning of \"factor of a polynomial\" and explain the meaning \"to factor a polynomial\"<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #95299 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=95299\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Remember, if you factor a number like 56, you simply break that number into it's product of prime numbers. So 56 factors to \r<BR>\n" );
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document.write( "56=2*2*2*7\r<BR>\n" );
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document.write( "So with polynomials, a similar thing occurs when you factor a polynomial\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x%2B6 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and 6 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient 6 to get 6. Now what two numbers multiply to 6 and add to the  middle coefficient 5? Let's list all of the factors of 6:\r<BR>\n" );
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document.write( "Factors of 6:\r<BR>\n" );
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document.write( "1,2,3,6\r<BR>\n" );
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document.write( "-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up and multiply to 6\r<BR>\n" );
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document.write( "1*6\r<BR>\n" );
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document.write( "2*3\r<BR>\n" );
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document.write( "(-1)*(-6)\r<BR>\n" );
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document.write( "(-2)*(-3)\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">6</TD><TD >1+6=7</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">3</TD><TD >2+3=5</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-6</TD><TD >-1+(-6)=-7</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-3</TD><TD >-2+(-3)=-5</TD></tr>\n" );
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document.write( "From this list we can see that 2 and 3 add up to 5 and multiply to 6\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x%2B6 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B3x ALIGN=MIDDLE  ALT=\"2x%2B3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B3x ALIGN=MIDDLE  ALT=\"2x%2B3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bhighlight%282x%2B3x%29%2B6 ALIGN=MIDDLE  ALT=\"x%5E2%2Bhighlight%282x%2B3x%29%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x%2B3x%2B6 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x%2B3x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B2x%29%2B%283x%2B6%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B2x%29%2B%283x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B2%29%2B3%28x%2B2%29 ALIGN=MIDDLE  ALT=\"x%28x%2B2%29%2B3%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x%2B6 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at the factorization, the terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the factors of the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x%2B6 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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